## Laws of the iterated logarithm for the range of random walks in two and three dimensions.(English)Zbl 1031.60031

Let $$(S_n)$$ be a random walk in $$Z^d$$, i.e. the sum of i.i.d. centered random variables $$X_i$$ taking values in $$Z^d$$. Let $$R_n$$ be the range of the random walk, i.e. $$R_n$$ is the number of distinct sites visited by $$S_0, S_1, \dots ,S_n$$. For $$d=3$$, under an additional assumption on the moments of $$X_1$$, the authors prove an almost sure invariance principle for $$R_n$$. As a corollary, they obtain laws of the iterated logarithm. For $$d=2$$, assuming that the two coordinates of $$X_1$$ are independent and bounded, they prove a law of the iterated logarithm for the range $$R_n$$.

### MSC:

 60G50 Sums of independent random variables; random walks 60F15 Strong limit theorems 60G17 Sample path properties 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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### References:

 [1] BASS, R. F. and KHOSHNEVISAN, D. (1992). Local times on curves and uniform invariance principles. Probab. Theory Related Fields 92 465-492. · Zbl 0767.60070 [2] BASS, R. F. and KUMAGAI, T. (2000). Laws of the iterated logarithm for some sy mmetric diffusion processes. Osaka J. Math. 37 625-650. · Zbl 0972.60007 [3] BENNETT, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33-45. · Zbl 0104.11905 [4] DONSKER, M. D. and VARADHAN, S. R. S. (1979). On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32 721-747. · Zbl 0418.60074 [5] DVORETZKY, A. and ERD OS, P. (1951). Some problems on random walk in space. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 353-367. Univ. California Press, Berkeley. · Zbl 0044.14001 [6] HAMANA, Y. (1995). On the multiple point range of three-dimensional random walks. Kobe J. Math. 12 95-122. · Zbl 0856.60073 [7] HAMANA, Y. (1997). The fluctuation result for the multiple point range of two-dimensional recurrent random walks. Ann. Probab. 25 598-639. · Zbl 0890.60066 [8] HAMANA, Y. (1998). An almost sure invariance principle for the range of random walks. Stochastic Process. Appl. 78 131-143. · Zbl 0934.60044 [9] HAMANA, Y. (2000). Personal communication. [10] HAMANA, Y. and KESTEN, H. (2001). A large-deviation result for the range of random walk and for the Wiener sausage. Probab. Theory Related Fields 120 183-208. · Zbl 1015.60092 [11] HAMANA, Y. and KESTEN, H. (2002). Large deviations for the range of an integer-valued random walk. Ann. Inst. H. Poincaré Probab. Statist. 38 17-58. · Zbl 1009.60084 [12] JAIN, N. C. and PRUITT, W. E. (1970). The range of recurrent random walk in the plane. Z. Wahrsch. Verw. Gebiete 16 279-292. · Zbl 0194.49205 [13] JAIN, N. C. and PRUITT, W. E. (1971). The range of transient random walk. J. Anal. Math. 24 369-393. · Zbl 0249.60038 [14] JAIN, N. C. and PRUITT, W. E. (1972). The law of the iterated logarithm for the range of random walk. Ann. Math. Statist. 43 1692-1697. · Zbl 0247.60042 [15] JAIN, N. C. and PRUITT, W. E. (1972). The range of random walk. Proc. Sixth Berkeley Sy mp. Math. Statist. Probab. 3 31-50. Univ. California Press, Berkeley. · Zbl 0247.60042 [16] JAIN, N. C. and PRUITT, W. E. (1974). Further limit theorems for the range of random walk. J. Anal. Math. 27 94-117. · Zbl 0293.60063 [17] KALLENBERG, O. (1997). Foundations of Modern Probability. Springer, New York. · Zbl 0892.60001 [18] LE GALL, J.-F. (1986). Propriétés d’intersection des marches aléatoires. I. Convergence vers le temps local d’intersection. Comm. Math. Phy s. 104 471-507. · Zbl 0609.60078 [19] LE GALL, J.-F. (1988). Fluctuation results for the Wiener sausage. Ann. Probab. 16 991-1018. · Zbl 0665.60080 [20] LE GALL, J.-F. (1994). Exponential moments for the renormalized self-intersection local time of planar Brownian motion. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1583 172-180. Springer, Berlin. · Zbl 0810.60078 [21] LE GALL, J.-F. and ROSEN, J. (1991). The range of stable random walks. Ann. Probab. 19 650-705. · Zbl 0729.60066 [22] SKOROHOD, A. V. (1965). Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA. · Zbl 0146.37701 [23] SPITZER, F. (1976). Principles of Random Walk. Springer, Berlin. · Zbl 0359.60003 [24] STORRS, CONNECTICUT 06269 E-MAIL: bass@math.uconn.edu RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES Ky OTO UNIVERSITY Ky OTO 606-8502 JAPAN E-MAIL: kumagai@kurims.ky oto-u.ac.jp
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